So I was recently attempting Question 6. off this GT example sheet (https://www.dpmms.cam.ac.uk/study/IA/Groups/2007-2008/g1.pdf) and I'm not too sure about a couple of the parts the question:
Let $C_{n}$ be the cyclic group with $n$ elements and $D_{2n}$ the group of symmetries of the regular $n$-gon. If $n$ is odd and $θ : D_{2n} → C_n$ is a homomorphism, show that $θ(g) = e$ for all $g ∈ D_{2n}$. What can you say if n is even? Find all the homomorphisms from $C_n$ to $C_m$.
First part:
I wrote $D_{2n} = \{ x^{i}y^{j} : i=0,1 \ \ 0 \leq j \leq n-1 \ xy=y^{-1}x \} $ and $C_{n}= \{ e, g, g^2, ... , g^{n-1} \} $. Let $\theta (x) = g^{i} ,\theta (y) = g^{j} $ Since the mapping is into the abelian cyclic group, clearly the images must commute. Using the fact $xy=y^{-1}x$, we have $ \theta (xy)= \theta (y^{-1}x)$
$ \Rightarrow \theta (x) \theta (y)= \theta (y^{-1}) \theta (x) = \theta (x) \theta (y) ^{-1} $
hence by cancellation laws, $\theta (y) = \theta(y)^{-1} $. My reasoning here is either $ \theta (y) = e' $ or $y^{n/2} $ if this exists. So clearly n is odd means we have $ \theta (y) = e' $ only. Also using $x^2=e$ I showed the similar result for x.
But the next part is where I am unsure- Find all the homomorphisms from $C_n$ to $C_m$.
The only way I can seem to directly relate it to the last part is find homomorphisms from $ C_{m} \rightarrow D_{2n} $ and then compose those with $ \theta $ from the last part, but since m and n may be different I dont know how to accomplish this.
So just by basic properties about homomorphisms, ignoring the dihedral groups, I managed to get down the following:
For phi is a homomorphism,
$ \phi (g^{-1})= \phi (g)^{-1} $. If $g \in C_{n}, h \in C_{m} $, $\phi(g)= h^i $ and so $\phi(g^{n-1})= h^{m-i} = h^{i(n-1)} \Rightarrow h^{in}= e', in= \lambda m$.
So if $n \not | m$ , $n | \lambda $ so $i=km, h^{i}=e'$ so $\phi$ takes everything to the identity.
But if $n|m$ we can have i not being a multiple of m, so $h^{i} \not= e' $.
So is my current argument correct? and is there a way to use part 1 for the last part/ have I missed out important details on the last part? Any help would be appreciated.